Many practical geomechanical problems require an estimate of the stresses in a formation beneath the earth's surface, whether the formation lies beneath a mass of land, water, or both land and water. Often, when time and costs are not a limiting factor, the stresses at a particular area of interest in a particular formation can be assessed using field stress measurement methods such as hydraulic fracturing methods, borehole ellipticity/breakout methods, formation integrity tests, and mini-frac tests, among other methods. Unfortunately, however, field stress measurements taken at one point in a formation can provide only a limited understanding, if any, of the stress distribution throughout the formation of interest. So, it has been difficult to determine, with reasonable accuracy and resolution, the stresses at other points in the formation, outside the area in which actual field stress measurements were obtained.
Field stress measurements taken in one region of a formation have been difficult to extrapolate to other points in the formation because the distribution of stresses in the formation can depend heavily on topography, far-field tectonic forces and local geologic history, among other factors. Consequently, before Applicants' invention, methods used to estimate the distribution of stresses in a formation have produced relatively inaccurate and unresolved stress values for other points in the formation outside the area in which actual field stress measurements were obtained.
One simplified approach that has been used previously, involves first determining a principal vertical stress, σvert, in which σvert is simply based on the weight of the overburden, or weight of rock, above the point of interest in the formation. Second, each principal horizontal stress, σhoriz-1 and σhoriz-2, is presumed to be proportional to σvert by a constant, but typically different, factor. For example, in the 1993 SPE paper (# 26074) entitled “Finite-Element Modeling of Depletion-Induced Reservoir Compaction and Surface Subsidence in the South Belridge Oil Field, California,” Hansen et al. suggested that the lesser of the two principal horizontal stresses equals 0.65 σvert, while the greater of the two principal horizontal stresses equals 1.20 σvert.
For purposes of determining a vertical stress with limited effort and expense, Hansen et al.'s approach provides a reasonable first order approximation for the formation's vertical stress, σvert. However, the proportionality assumes that for any given formation, a horizontal stress is consistently related to the formation's vertical stress, where the overburden weight (used to determine σvert) is based on an average rock density for a single point or area in the formation. This can be acceptable for a simple first order approximation. However, such an approximation implicitly neglects variability in rock properties and topography throughout a formation, frequently found in the formations of interest, and past geologic processes (e.g., deposition, erosion, tectonics, etc.) that can contribute to a formation's present-day stress distribution. So, substantial variations in the formation's stress distribution, arising from variability in rock properties and geologic processes leading to the formation's creation, are not accounted for using a formation stress approximation method like the one disclosed by Hansen et al.
Consequently, even if the initial approximation of σvert is a reasonable one, a simplified approximation method can produce an over-simplified model of a formation's stress distribution, particularly with respect to the principal horizontal stresses. Such an over-simplified model of a formation's stress distribution, like that produced using the Hansen et al. assumptions, for instance, can produce a relatively less resolved and less accurate estimate of stresses at any point in the formation. In turn, the over-simplified model tends to be less helpful in predicting the effect, if any, man-induced stresses (e.g., injecting a fluid at high pressure, depleting formation fluids, formation fracturing, explosion, etc.) might have on different area(s) of interest in the formation.
Another conventional approach, discussed in Blanton et al. (“Stress Magnitudes from Logs: Effects of Tectonic Strains and Temperature” SPE Reservoir Eval. & Eng. 2:1:February 1999 and referencing Gatens et al. “In-Situ Stress Tests and Acoustic Logs Determine Mechanical Properties and Stress Profiles in the Devonian Shales” SPE 18523; 1990), is to first determine a σvert based on present-day overburden weight. Then the corresponding σhoriz-2 is estimated by Equation (1), using present-day Poisson ratio values and σvert.
                              σ                      horiz            -            2                          =                                                            v                Present                                            1                -                                  v                  Present                                                      ⁢                          (                                                σ                  vert                                -                                                      α                    p                                    ⁢                  p                                            )                                +                                    α              p                        ⁢            p                                              (        1        )            where
vpresent is a measured present-day Poisson ratio value (dimensionless)
σhoriz-2 is a minimum principal horizontal stress (psi)
σvert is a principal vertical stress (psi)
αp is Biot's poroelastic constant (dimensionless)
p is pore pressure (psi)
Note: Eq. (1) as shown has been amended to conform with the nomenclature of the present application.
Well logs are used to produce a set of present-day Poisson ratio, vPresent, values as a function of depth. Eq. (1) is then used to calculate σhoriz-2 values as a function of depth for a location where calibrated data is available.
Whether calculated according to Hansen et al. (where σhoriz-2 and σhoriz-1 are multipliers of σvert) or by Eq. (1), the actual stress measurements for one location are then used to assess a formation's present-day stress distribution by simply extrapolating known, present-day stress measurements from one location to another distant location one-dimensionally. That is, stresses, whether vertical or horizontal, at any given depth in the formation are assumed to be a function of depth from the surface and extending substantially uniformly, radially outward within the radial plane from one area, where actual field stress data is available, to any other point in the location, where no such data is available.
In more pictorial terms, this simplified approach to modeling a formation's stress distribution assumes a formation is depicted, in effect, by an infinite number of spoked wheels, one atop the other. Meanwhile, actual σvert is determined according to changes in depth, and hence, horizontal stresses are assumed as “known” at each wheel's hub. In turn, these vertical and horizontal stresses are then extrapolated radially outward, along any spoke (also assuming an infinite number of spokes around each “hub” area) to any other point of interest in the formation.
And to the extent field data is available at two or more separate areas of a formation, then a formation model, based on this simplified approach, could be better refined by simply taking some intermediate value (i.e., interpolating) between different stress results obtained for the point(s) of interest, as produced by using multiple sets of stress data taken/obtained for multiple locations throughout the formation and producing corresponding sets of overlapping spoked-wheel stacks for depicting the formation. And again, to the extent there is no convergence for the spokes in the same radial plane extending out from the independent hub data sets to where no stress data is available, then an intermediate or interpolated stress value is typically generated, accordingly.
Of course, taking and/or obtaining field stress data at strategic and multiple locations throughout a formation, to produce the desired stress analysis, is both time consuming and costly, if not sometimes prohibitive for a lack of time, money or both. Consequently, it would be preferable to have a method for calibrating a model of a formation's stress distribution that more accurately reflects the formation's actual, present-day stress distribution for the intended stress analysis, and more preferably, have a method that can produce such a model using stress data from a single area of a formation. For example, such a calibration procedure should develop, within the desired degree of certainty, a model of the formation's stress distribution that more accurately captures the 3-dimensional stress variations that typically exist in a formation.
Consequently, a different approach is required for developing a truer model of a formation's stress distribution from stress data at one or more location(s) versus developing an artificial 3-D construct, like that used by conventional methods. Again, such conventional methods basically assume that principal stresses at one location can be extended one-dimensionally, radially outward (i.e., extrapolated) to any other location, where no such data is available, while effectively neglecting rock property variations and/or geohistorical effects on a formation's present-day stress distribution, whether in a virgin (i.e., before a man-induced, stress-altering event occurs in the formation) or non-virgin state. Moreover, these three-dimensional stress variations serve to redistribute the variable gravitational loads caused by topographic relief, which have been ignored in the conventional methods discussed above. While ignoring topographic relief can sometimes produce an adequate model for certain formations, there is often a need for a better characterization of the stress distribution in a formation as a whole.
Therefore, despite the reasonable correlation between σvert and the weight of a formation's rock, certain subsequent assumptions can produce a less resolved and less accurate estimate of the formation's stress distribution suitable for performing the desired formation stress analysis. For example, assumptions such as: (1) that σvert is correlated to each principal horizontal stress, σhoriz-1 and σhoriz-2, by a predetermined constant factor (e.g., 1.20 and 0.65, respectively) or by Eq. (1) and/or (2) that the formation's rock properties are substantially homogeneous throughout the formation, can significantly reduce the resolution and accuracy of a stress distribution model for a formation based on such assumptions. Accordingly, there is a need for an improved method of determining that a model of a formation's stress distribution is suitably calibrated to the formation of interest, so that the desired stress analysis at any point in the formation can be performed with improved accuracy and/or resolution versus more simplified formation modeling methods previously used.